SolidStatePhysicsPPT學(xué)習(xí)課件PPT課件

1、Chapter 6: Free Electron Fermi Gas第1頁/共32頁123返回退出自由電子氣模型自由電子氣模型一、價(jià)電子與傳導(dǎo)電子一、價(jià)電子與傳導(dǎo)電子自由Na原子的的電子組態(tài)電子組態(tài)16223221SPSS價(jià)電子軌道半徑0.19nm價(jià)電子決定了元素的大多數(shù)化學(xué)性質(zhì)第2頁/共32頁123返回退出自由電子氣模型自由電子氣模型一、價(jià)電子與傳導(dǎo)電子一、價(jià)電子與傳導(dǎo)電子 價(jià)電子 離子實(shí)間距0.37nm 傳導(dǎo)電子傳導(dǎo)電子決定了金屬的大多數(shù)特性第3頁/共32頁123返回退出自由電子氣模型自由電子氣模型二、德魯特模型二、德魯特模型1900年 德魯特自由電子氣模型第4頁/共32頁123返回退出自

2、由電子氣模型自由電子氣模型二、德魯特模型二、德魯特模型自由電子氣系統(tǒng)32910mMZNVNnmA理想氣體系統(tǒng) 32510m濃度約第5頁/共32頁123返回退出自由電子氣模型自由電子氣模型二、德魯特模型二、德魯特模型自由電子氣系統(tǒng)理想氣體系統(tǒng) 濃度大 濃度小電子氣帶電理想氣體分子電中性第6頁/共32頁123返回退出自由電子氣模型自由電子氣模型二、德魯特模型二、德魯特模型電子的無軌熱運(yùn)動(dòng)和漂移運(yùn)動(dòng)的疊加mne2第7頁/共32頁123返回退出自由電子氣模型自由電子氣模型二、德魯特模型二、德魯特模型電子運(yùn)動(dòng)方程穩(wěn)態(tài)解電流密度電導(dǎo)率 mne2VmEedtvdmEmeVdEmenVnejd2第8頁/共32

3、頁123返回退出自由電子氣模型自由電子氣模型三、成功與失敗三、成功與失敗電電子子濃濃度度 n: 32928/1010m室室溫溫下下 : m810求得求得 馳豫時(shí)間馳豫時(shí)間 : s14151010 電子平均速度電子平均速度 v: sm/105求得求得 平均自由程平均自由程 v: nm101 . 0 似乎自恰性相當(dāng)好似乎自恰性相當(dāng)好實(shí)實(shí) 驗(yàn)驗(yàn) 結(jié)結(jié) 果果 ： : 100 nm 以以 上上 此此 外外 ： 電電 子子 比比 熱熱 、 磁磁 化化 率率 等等 也也 遇遇 到到 了了 困困 難難 。第9頁/共32頁123返回退出自由電子氣模型自由電子氣模型三、成功與失敗三、成功與失敗 電子平均自由程：電

4、阻率： 電子對比熱貢獻(xiàn)： 自由電子氣 模型計(jì)算實(shí)驗(yàn)結(jié)果100nm0.1-1.0nm較小僅為計(jì)算值的1/200較大怎么辦？TT第10頁/共32頁Chapter 6: Free Electron Fermi Gas The alkali metals: Li, Na, K, Cs, Rb. Na, 3s conduction band. A monovalent crystal which contains N atoms will have N conduction electrons and N positive ion cores. The ion cores fill only about

5、 15 percent of the volume of a sodium crystal ( Na+, R=0.98A)第11頁/共32頁Chapter 6: Free Electron Fermi Gas The classical theory of the free electron model: Ohms law, the relation between the electrical and thermal conductivity; the heat capacity and the magnetic susceptibility (Maxwell distribution fu

6、nction). Experiments: a conduction electron in a metal can move freely in a straight path over many atomic distances.第12頁/共32頁Chapter 6: Free Electron Fermi Gas Why is condensed matter so transparent to conduction electrons? (1) A conduction electron is not deflected by ion cores arranged on a perio

7、dic lattice; (2) A conduction electron is only infrequently by other conduction electrons. (Pauli principle). A Free electron Fermi gas: a gas of free electrons subject to the Pauli principle)第13頁/共32頁1. Energy Levels in One Dimension Consider a free electron gas in one dimension, taking account of

8、quantum theory and of the Pauli principle. An electron of mass m is confined to a length L by infinite barriers. The Schrodinger equation (-2/2m)d2n/dx2=n n , where n is the energy of the electron in the orbital.第14頁/共32頁1. Energy Levels in One Dimension The Schrodinger equation (-2/2m)d2n/dx2=n n ,

9、 where n is the energy of the electron in the orbital. The boundary conditions are n (0)=0; n (L)=0. Then n (x)=Asin(2 x/ n); n n/2=L,where A is a constant. 第15頁/共32頁1. Energy Levels in One Dimension第16頁/共32頁1. Energy Levels in One Dimension The energy n of the electron is given by n = (2/2m)(n /L)2

10、 . In a linear solid the quantum numbers of a conduction electron orbital are n and ms(=1/2). The number of orbitals with the same energy is called the degeneracy. Let nF is the topmost filled energy level, the condition 2 nF=N determines nF.第17頁/共32頁1. Energy Levels in One Dimension第18頁/共32頁1. Ener

11、gy Levels in One Dimension The Fermi energy is defined as the energy of the topmost filled level in the ground state of the N electron system. ThusF = (2/2m)(nF /L)2 = (2/2m)(N /2L)2 .第19頁/共32頁2. Effect of Temperature on the Fermi-Dirac Distribution The ground state is the state of the N electron sy

12、stem at absolute zero. The Fermi-Dirac distribution gives the probability that an orbital at energy will be occupied in an ideal electron gas in thermal equilibrium:第20頁/共32頁2. Effect of Temperature on the Fermi-Dirac Distribution第21頁/共32頁2. Effect of Temperature on the Fermi-Dirac Distribution The

13、quantity is a function of the temperature and is the chemical potential. At absolute zero =F (the energy of the topmost filled orbital at T=0). At all temperatures f() is equal to 1/2 when =; When -kBT, f() =exp(- )/ kBT (Boltzmann or Maxwell distribution)第22頁/共32頁3. Free Electron Gas in 3D The free

14、-particle Schrodinger equation in 3D is第23頁/共32頁3. Free Electron Gas in 3D The periodic boundary conditions: (x+L,y,z)= (x,y,z), (x,y+L,z)= (x,y,z), (x,y,z+L)= (x,y,z). Wavefunctions satisfying the free-particle Schro-dinger equation and the periodicity condition are of the form of a travelling plan

15、e wave: k(r)=exp(ikr),第24頁/共32頁3. Free Electron Gas in 3D The wavevector k satisfy kx=0; 2/L; 4/L; , and similarly for kx and kz . Any component of k is of the form 2n /L, where n is an integer. The energy k of the orbital with wavevector k: k=2k2/2m= (2/2m)(k x2 + k y2 + k z2 ).第25頁/共32頁3. Free Ele

16、ctron Gas in 3D p k(r) =-i k(r) = k k(r), so that the plane wave k(r) is an eigenfunction of the linear momentum with the eigenvalue k; The particle velocity in the orbital k is given by v=k/m. In the ground state of a system of N free electrons the occupied orbitals may be represented as points ins

17、ide a sphere in k space.第26頁/共32頁3. Free Electron Gas in 3D 第27頁/共32頁3. Free Electron Gas in 3D The energy at the surface of the sphere is the Fermi energy. The wavevectors at the Fermi surface have a magnitude kF such that: k=2kF2/2m. In the sphere of volume 4kF3/3 the total number of orbitals is 2

18、(L/2 )3(4kF3/3 )=N,第28頁/共32頁3. Free Electron Gas in 3D 2(L/2 )3(4kF3/3 )=N, Then kF=(32N/V)1/3, which depends only on the particle concentration. The Fermi energy is F=(2/2m) (32N/V)2/3. The Fermi velocity is vF=(kF/m) =(/m) (32N/V)1/3. Fermi temperature TF= F/ kB第29頁/共32頁3. Free Electron Gas in 3D 第30頁/共32頁3. Free Electron Gas in 3D The total number of orbitals of energy : N=(V/32)(2m/ 2)3/2, so that the density of states